group-theory

Quadratics, cubics and quartics all have formulas in radicals. The general fifth-degree equation has none, and not because nobody clever enough has tried. This lesson explains how permuting the roots turns an algebra question into a group question, and why the answer changes at degree five when a group of order 60 refuses to break down.

Group theory earns its keep when a symmetry argument settles a physical question no calculation was going to answer. This lesson covers group actions and counting up to symmetry, why crystals cannot have five-fold rotation and what happened when one did, Noether's link between symmetry and conservation, and how the same idea is built into modern networks.

Cosets cut a group into equal-sized blocks, and one counting argument then forces every subgroup's size to divide the group's. This lesson proves that, derives Fermat's little theorem from it as a corollary, and introduces the maps that let you compare two groups and say precisely when they are the same object.

A group is a set with one operation obeying four rules, and that deliberate poverty is the point: anything proved about groups holds for rotations, permutations, integers, XOR and Rubik's cube at once. This lesson builds the definition from symmetry, works through the symmetries of a square, and shows where groups already sit in code.